Square Root of -90
2026-02-28 10:37 Diff
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Last updated on August 5, 2025

When a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The concept of square roots extends into complex numbers when dealing with negative values. Here, we will discuss the square root of -90.

What is the Square Root of -90?

The square root is the inverse of squaring a number. Since -90 is a negative number, its square root is not a real number. Instead, it is a complex number. The square root of -90 is expressed in both radical and exponential form. In radical form, it is expressed as √(-90), whereas in exponential form, it is (-90)^(1/2). The square root of -90 is an imaginary number, which can be expressed as 3√10i, where i is the imaginary unit defined as √(-1).

Finding the Square Root of -90

The square root of a negative number involves the imaginary unit i. For positive numbers, methods like prime factorization, long division, and approximation are used. However, for negative numbers, we focus on expressing them in terms of i. Let us explore this concept further: Imaginary unit i Expressing negative roots in terms of i Simplifying under the radical

Square Root of -90 by Expressing in Terms of i

To express the square root of a negative number using imaginary numbers, we follow these steps:

Step 1: Express -90 as -1 × 90.

Step 2: Recognize that the square root of -1 is i, the imaginary unit. Thus, √(-1) = i.

Step 3: Write √(-90) as √(-1 × 90) = √(-1) × √90 = i√90.

Step 4: Simplify √90 by finding its prime factors: 90 = 2 × 3^2 × 5. Thus, √90 = √(3^2 × 10) = 3√10.

Step 5: Combine the expressions to get √(-90) = 3√10i.

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Understanding Complex Numbers in the Context of Square Roots

Complex numbers, which include real and imaginary parts, are essential for understanding square roots of negative numbers. The square root of -90 is purely imaginary, expressed as 3√10i. Real numbers cannot yield a negative square when squared; hence, imaginary numbers are used. Let's elaborate on this: Real part: zero for purely imaginary numbers Imaginary part: derived from the radical simplification Application of complex numbers in various fields

Applications and Importance of Imaginary Numbers

Imaginary numbers extend the real number system to solve equations lacking real solutions. Their applications include electrical engineering, quantum physics, and complex number theory. Understanding square roots of negative numbers is foundational in these fields: Electrical circuits: alternating current representations Quantum mechanics: wave functions and probabilities Complex analysis: advanced calculus and engineering

Common Mistakes and How to Avoid Them with the Square Root of -90

Mistakes frequently occur when dealing with square roots of negative numbers due to misunderstandings about imaginary units and complex numbers. Let us explore common errors and how to avoid them:

Problem 1

Can you help Max find the magnitude of a complex number if it is given as 5 + √(-90)?

Okay, lets begin

The magnitude of the complex number is approximately 20.12.

Explanation

The magnitude of a complex number a + bi is √(a^2 + b^2). For 5 + √(-90), express √(-90) as 3√10i. Thus, we have 5 + 3√10i. Calculate: √(5^2 + (3√10)^2) = √(25 + 90) = √115 ≈ 20.12.

Well explained 👍

Problem 2

If the square root of -90 is added to a real number 10, what is the result in complex number form?

Okay, lets begin

The result is 10 + 3√10i.

Explanation

The square root of -90 is expressed as 3√10i. Adding this to the real number 10 gives the complex number 10 + 3√10i, with 10 as the real part and 3√10i as the imaginary part.

Well explained 👍

Problem 3

Calculate the result of squaring the square root of -90.

Okay, lets begin

The result is -90.

Explanation

Squaring the square root of -90, which is 3√10i, results in (3√10i)^2 = (3√10)^2 × i^2 = 90 × (-1) = -90, since i^2 = -1.

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Problem 4

What is the conjugate of the complex number 7 + √(-90)?

Okay, lets begin

The conjugate is 7 - 3√10i.

Explanation

The conjugate of a complex number a + bi is a - bi. For 7 + √(-90), express √(-90) as 3√10i. Thus, the conjugate is 7 - 3√10i.

Well explained 👍

Problem 5

Find the imaginary part of the complex number formed by adding 5 to the square root of -90.

Okay, lets begin

The imaginary part is 3√10i.

Explanation

Form the complex number by adding 5 to the square root of -90, which is 3√10i. Therefore, the complex number is 5 + 3√10i, with the imaginary part being 3√10i.

Well explained 👍

FAQ on Square Root of -90

1.What is √(-90) in its simplest form?

The simplest form of √(-90) is 3√10i, as it involves expressing the negative root in terms of the imaginary unit i and simplifying the radical expression.

2.What are the applications of imaginary numbers like √(-90)?

Imaginary numbers are used in electrical engineering (AC circuits), quantum physics (wave functions), and complex number analysis, providing solutions for equations with no real roots.

3.Why can't the square root of -90 be a real number?

A real number squared gives a non-negative result. Since squaring any real number can't yield -90, its square root must involve the imaginary unit i.

4.Can the square root of a negative number be simplified without using i?

No, the square root of a negative number inherently involves i, the imaginary unit, which allows for the expression and calculation of non-real square roots.

5.What is the significance of the imaginary unit i?

The imaginary unit i, defined as √(-1), is crucial in extending the real number system to complex numbers, enabling the solution of equations lacking real solutions.

Important Glossaries for the Square Root of -90

  • Square root: The inverse operation of squaring a number. For negative numbers, it involves imaginary numbers. Example: The square root of -90 is 3√10i.
     
  • Imaginary number: A number in the form of a real number multiplied by the imaginary unit i, representing the square root of negative values. Example: 3i is an imaginary number.
     
  • Complex number: A number comprising a real part and an imaginary part, expressed as a + bi. Example: 5 + 3√10i is a complex number.
     
  • Imaginary unit (i): A mathematical constant defined as √(-1), enabling calculations with negative square roots.
     
  • Conjugate: For a complex number a + bi, the conjugate is a - bi, important in complex arithmetic. Example: The conjugate of 7 + 3√10i is 7 - 3√10i.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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: He loves to play the quiz with kids through algebra to make kids love it.